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Further results on iterative methods for computing generalized inverses. (English) Zbl 1190.65088

Let $A$ be a complex Banach algebra with unit 1; $𝒳,𝒴$ two complex Banach spaces and $ℬ\left(𝒳,𝒴\right)$ the set of all bounded linear operators from $𝒳$ to $𝒴$. The main theorem is stated as follows: Define the sequence

${X}_{k}={X}_{k-1}+\beta Y\left({I}_{𝒴}-A{X}_{k-1}\right),\phantom{\rule{1.em}{0ex}}k=1,2,\cdots ,$

where $\beta \in ℂ\setminus \left\{0\right\}$ and ${X}_{0}\in ℬ\left(𝒴,𝒳\right)$ with $Y\ne YA{X}_{0}·$ Then the above iteration converges if and only if $\rho \left({I}_{𝒳}-\beta YA\right)<1,$ equivalently, $\rho \left({I}_{𝒴}-\beta AY\right)<1·$

In this case suppose now $\left(\rho \left({I}_{𝒳}-\beta YA\right)<1$ and that $T,S$ are closed subspaces of $𝒳,𝒴·$ If moreover $ℛ\left(Y\right)=T$,   $𝒩\left(Y\right)=S\phantom{\rule{1.em}{0ex}}$and $ℛ\left({X}_{0}\right)\subset T$, then ${A}_{T,S}^{\left(2\right)}$ exists and $\left\{{X}_{k}\right\}$ converges to ${A}_{T,S}^{\left(2\right)}$ and if $q=min\left(\parallel {I}_{𝒳}-\beta YA\parallel ,\parallel {I}_{𝒴}-\beta AY\parallel \right)<1$:

$\parallel {A}_{T,S}^{\left(2\right)}-{X}_{k}\parallel \le \frac{|\beta |{q}^{k}}{1-q}\parallel Y\parallel \parallel {I}_{𝒴}-A{X}_{0}\parallel ·$

Here ${A}_{T,S}^{\left(2\right)}$ denotes the generalized inverse. Several following theorems yield variants of the above theorem which state conditions under which the iterative procedures approximate the generalized inverse. The next section entitled “The generalized Drazin inverse of Banach algebra elements” defines an iteration approximating this inverse and different conditions under which this sequence converges to the Drazin inverse. Finally, the last section is devoted to a numerical example in which $A\in {ℂ}^{5×4}·$

##### MSC:
 65J10 Equations with linear operators (numerical methods) 47A05 General theory of linear operators 15A09 Matrix inversion, generalized inverses 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra)